The aforelinked Tweetbit is the first full iPhone Twitter client that includes its own support for push notifications, but there are at least two other apps that focus just on push notifications, while integrating with existing Twitter clients.

Boxcar costs $2, and checks your Twitter account every two minutes. It sends push notifications for direct messages and/or mentions. For mentions, it lets you choose between opening them in Tweetie or Twitterrific; for DMs, though, Boxcar shows them in its own interface.